The Zak transform: a framework for quantum computation with the Gottesman-Kitaev-Preskill code

TL;DR

This paper explores the use of the Zak transform in quantum computation with the GKP code, providing a new framework for understanding error correction and subsystem decomposition in bosonic modes.

Contribution

It introduces a novel Zak transform-based framework and a new bosonic subsystem decomposition for GKP codes, enhancing understanding of error correction.

Findings

Zak transform naturally describes GKP error correction

New modular variable subsystem decomposition constructed

Trace over gauge mode yields logical qubit states

Abstract

The Gottesman-Kitaev-Preskill (GKP) code encodes a qubit into a bosonic mode using periodic wavefunctions. This periodicity makes the GKP code a natural setting for the Zak transform, which is tailor-made to provide a simple description for periodic functions. We review the Zak transform and its connection to a Zak basis of states in Hilbert space, decompose the shift operators that underpin the stabilizers and the correctable errors, and we find that Zak transforms of the position wavefunction appear naturally in GKP error correction. We construct a new bosonic subsystem decomposition (SSD) -- the modular variable SSD -- by dividing a mode's Hilbert space, expressed in the Zak basis, into that of a virtual qubit and a virtual gauge mode. Tracing over the gauge mode gives a logical-qubit state, and preceding the trace with a particular logical-gauge interaction gives a different logical state -- that associated to GKP error correction.